Shortest String That Contains Three Strings

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Given three strings a, b, and c, your task is to find a string that has the minimum length and contains all three strings as substrings.

If there are multiple such strings, return the lexicographically smallest one.

Return a string denoting the answer to the problem.

Notes

A string a is lexicographically smaller than a string b (of the same length) if in the first position where a and b differ, string a has a letter that appears earlier in the alphabet than the corresponding letter in b.
A substring is a contiguous sequence of characters within a string.

Example 1:

Input: a = "abc", b = "bca", c = "aaa"
Output: "aaabca"
Explanation:  We show that "aaabca" contains all the given strings: a = ans[2...4], b = ans[3..5], c = ans[0..2]. It can be shown that the length of the resulting string would be at least 6 and "aaabca" is the lexicographically smallest one.

Example 2:

Input: a = "ab", b = "ba", c = "aba"
Output: "aba"
Explanation: We show that the string "aba" contains all the given strings: a = ans[0..1], b = ans[1..2], c = ans[0..2]. Since the length of c is 3, the length of the resulting string would be at least 3. It can be shown that "aba" is the lexicographically smallest one.

Constraints:

1 <= a.length, b.length, c.length <= 100
a, b, c consist only of lowercase English letters.

Try coding it in LeetCode

Solution

Clarifying Questions

When you get asked this question in a real-life environment, it will often be ambiguous (especially at FAANG). Make sure to ask these questions in that case:

What is the maximum length of each of the three input strings?
Can any of the input strings be empty or null?
If multiple shortest strings exist, is any one acceptable or is there a specific criteria for choosing among them?
Are the input strings guaranteed to contain only ASCII characters, or might they contain Unicode characters?
Do the three input strings need to be distinct (i.e., different content), or can they be identical?

Brute Force Solution

Approach

The brute force method for finding the shortest string that contains three other strings involves trying every single possible arrangement of those three strings. We consider all possible ways the strings could overlap and combine them to see which combined string is the shortest. This is done by exhaustively checking all options.

Here's how the algorithm would work step-by-step:

First, consider all the possible orders in which you can arrange the three strings. For example, string A then string B then string C, or string B then string C then string A, and so on.
For each of these orderings, try to combine the strings in every possible way. This means letting them be completely separate, or having them overlap partially, or even having one string completely contained within another.
When checking overlaps, shift the strings relative to each other in small increments to see if the ending of one string matches the beginning of the next.
Calculate the length of the resulting combined string for each arrangement and overlap.
Keep track of the shortest combined string you find during this whole process.
After checking all orderings and all possible overlaps within each ordering, the shortest combined string that you saved is the answer.

Code Implementation

def shortest_string_containing_three_strings(string1, string2, string3):
    from itertools import permutations

    strings = [string1, string2, string3]
    shortest_combined_string = None

    # Iterate through all possible permutations of the input strings
    for permutation in permutations(strings):
        
        first_string = permutation[0]
        second_string = permutation[1]
        third_string = permutation[2]

        # Try all possible overlaps between the first and second string
        for first_overlap in range(-len(first_string) + 1, len(second_string)):
            combined_first_second = combine_strings(first_string, second_string, first_overlap)
            
            # Then, try all possible overlaps between the combined first two strings, and the third string
            for second_overlap in range(-len(combined_first_second) + 1, len(third_string)):                
                combined_all_strings = combine_strings(combined_first_second, third_string, second_overlap)

                # Keep track of the shortest combination
                if shortest_combined_string is None or len(combined_all_strings) < len(shortest_combined_string):
                    shortest_combined_string = combined_all_strings

    return shortest_combined_string

def combine_strings(first_string, second_string, overlap):
    if overlap >= 0:
        # Second string overlaps to the right of the first string
        prefix = first_string + second_string[overlap:]

    else:
        # Second string overlaps to the left of the first string
        prefix = second_string[:abs(overlap)] + first_string

    return prefix

Big(O) Analysis

Time Complexity

O(n^3) – The algorithm considers all permutations of the three input strings, which is a constant factor (3! = 6). For each permutation, it tries to merge the strings with all possible overlaps. Overlap checking involves comparing substrings of the strings. Let n be the maximum length of any of the three input strings. For each pair of strings within a permutation, we shift one string relative to the other up to n positions. For each shift, a substring comparison of length at most n is performed. Since we have three strings, we perform this shift and comparison process at most twice per permutation, resulting in a time complexity of O(n^2) per pair and O(n^3) in total since we need to do all string concatenations. Therefore, the overall time complexity is O(n^3).

Space Complexity

O(1) – The algorithm primarily uses a few variables to store the current arrangement of strings, the lengths of strings, and the shortest combined string found so far. The space used by these variables is constant and does not depend on the input string lengths. There are no auxiliary data structures like lists, hash maps, or recursion that grow with the input size (N, where N is the combined length of the three input strings). Therefore, the space complexity is O(1).

Optimal Solution

Approach

The most efficient approach involves cleverly combining the three input strings to find the shortest possible result. We explore different orderings and look for significant overlaps between the strings to minimize the total length. By strategically merging strings, we avoid unnecessary duplication and achieve optimal shortness.

Here's how the algorithm would work step-by-step:

Consider all six possible orderings of the three input strings (e.g., string1-string2-string3, string1-string3-string2, and so on).
For each ordering, try to merge the strings together. Start by checking if the end of the first string matches the beginning of the second string, and if so, combine them to avoid duplication.
Repeat this merging process between the second and third strings in the current ordering, again looking for overlaps.
Calculate the length of the combined string for each of the six orderings.
Return the shortest combined string that you found from among all the orderings.

Code Implementation

def shortest_string_that_contains_three_strings(string1, string2, string3):
    import itertools

    def merge_strings(first_string, second_string):
        # Find the maximum overlap between two strings.
        for overlap_length in range(min(len(first_string), len(second_string)), 0, -1):
            if first_string[-overlap_length:] == second_string[:overlap_length]:
                return first_string + second_string[overlap_length:]
        return first_string + second_string

    permutations = list(itertools.permutations([string1, string2, string3]))
    shortest_combined_string = None

    for current_permutation in permutations:
        # Iterate through all possible orderings to find shortest.
        first_merge = merge_strings(current_permutation[0], current_permutation[1])
        combined_string = merge_strings(first_merge, current_permutation[2])

        if shortest_combined_string is None or len(combined_string) < len(shortest_combined_string):
            # Update shortest string if current one is shorter.
            shortest_combined_string = combined_string

    return shortest_combined_string

Big(O) Analysis

Time Complexity

O(n) – The algorithm considers 6 fixed permutations of the input strings, a constant number. For each permutation, the merging process involves comparing prefixes and suffixes of the strings to find overlaps. The length of the strings determines the size of these prefixes and suffixes to compare, which is at most 'n' (the maximum length of any one of the three strings). The overlap check within each permutation takes O(n) time. Thus, we perform O(n) work a constant (6) number of times, so the overall time complexity is O(n).

Space Complexity

O(N) – The space complexity is determined by the temporary strings created when merging the input strings in different orders. In the worst-case scenario, when there are no overlaps between the strings, the merged strings will have lengths close to the sum of the lengths of the three input strings. Therefore, we need extra space proportional to the length of combined strings, where N represents the total number of characters in the three input strings. Hence, the space complexity is O(N).

Edge Cases

Case	How to Handle
One or more input strings are null or empty.	Return an empty string or throw an IllegalArgumentException to handle null or empty input strings appropriately.
All three input strings are identical.	Return the input string since it trivially contains all three strings.
Two of the three input strings are identical.	Check for overlaps of identical strings to efficiently construct the shortest string.
One input string is a substring of another.	Prioritize the longer string and intelligently incorporate the third string while minimizing length.
No overlap exists between any two input strings.	Concatenate all three strings in all 6 possible permutations and return the shortest one.
The shortest string is not unique (multiple solutions with the same length).	Return any one of the shortest strings as the problem statement only asks for one solution.
Input strings are extremely long, potentially leading to memory issues.	Employ a more memory efficient string comparison algorithm or a streaming approach if possible.
Overlapping characters in a way that creating optimal merge of 2 strings is complex.	Implement an efficient longest common substring/prefix algorithm for effective overlap detection.

One or more input strings are null or empty.

How to Handle:

Return an empty string or throw an IllegalArgumentException to handle null or empty input strings appropriately.

All three input strings are identical.

How to Handle:

Return the input string since it trivially contains all three strings.

Two of the three input strings are identical.

How to Handle:

Check for overlaps of identical strings to efficiently construct the shortest string.

One input string is a substring of another.

How to Handle:

Prioritize the longer string and intelligently incorporate the third string while minimizing length.

No overlap exists between any two input strings.

How to Handle:

Concatenate all three strings in all 6 possible permutations and return the shortest one.

The shortest string is not unique (multiple solutions with the same length).

How to Handle:

Return any one of the shortest strings as the problem statement only asks for one solution.

Input strings are extremely long, potentially leading to memory issues.

How to Handle:

Employ a more memory efficient string comparison algorithm or a streaming approach if possible.

Overlapping characters in a way that creating optimal merge of 2 strings is complex.

How to Handle:

Implement an efficient longest common substring/prefix algorithm for effective overlap detection.